Thank you for your question today, I look forward to assisting you. I have nearly 20 years of legal experience in various disciplines, including JAG.
They can't give you a dishonorable discharge without court martialing you. Now, can they post a warrant for your arrest for AWOL? Yes, they certainly can.
Then one of two things happens. First, they just leave the warrant out there and leave you on their rolls. Eventually, you'll get pulled over some day, perhaps as many as 10 years from now when you've forgotten about this issue, and you'll end up arrested. That happens more than I like to think about. Second, the commander decides not to do that, and instead processes you for an administrative separation, which they can do without a court martial. The lowest discharge status you can get there is an Other than Honorable...not a dishonorable.
The second result is the more likely result, if you plan on leaving and never going back.
The issues with your job, while I certainly understand them (I am a reserve officer, so I understand having a full time life), are not a legal defense against the AWOL. USERRA protects your job and forces employers to respect military duty, so you'll have no legal defense to missing the AT.
If you can make it, at all, you need to do this AT.